Solve frac{sqrt{5}+1}{sqrt{5}-1}+frac{sqrt{5}-1}{sqrt{5}}-2sqrt{(1/5)^0-1/138}

$$\frac { \sqrt { 5 } + 1 } { \sqrt { 5 } - 1 } + \frac { \sqrt { 5 } - 1 } { \sqrt { 5 } } - 2 \sqrt { ( \frac { 1 } { 5 } ) ^ { 0 } - \frac { 1 } { 138 } }$$

Evaluate

$$\frac{3\sqrt{5}}{10}-\frac{\sqrt{18906}}{69}+\frac{5}{2}\approx 1.178079945$$

Short Solution Steps

Rationalize the denominator of \frac{\sqrt{5}+1}{\sqrt{5}-1} by multiplying numerator and denominator by \sqrt{5}+1.

$$\frac{\left(\sqrt{5}+1\right)\left(\sqrt{5}+1\right)}{\left(\sqrt{5}-1\right)\left(\sqrt{5}+1\right)}+\frac{\sqrt{5}-1}{\sqrt{5}}-2\sqrt{\left(\frac{1}{5}\right)^{0}-\frac{1}{138}}$$

Consider \left(\sqrt{5}-1\right)\left(\sqrt{5}+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(\sqrt{5}+1\right)\left(\sqrt{5}+1\right)}{\left(\sqrt{5}\right)^{2}-1^{2}}+\frac{\sqrt{5}-1}{\sqrt{5}}-2\sqrt{\left(\frac{1}{5}\right)^{0}-\frac{1}{138}}$$

Square \sqrt{5}. Square 1.

$$\frac{\left(\sqrt{5}+1\right)\left(\sqrt{5}+1\right)}{5-1}+\frac{\sqrt{5}-1}{\sqrt{5}}-2\sqrt{\left(\frac{1}{5}\right)^{0}-\frac{1}{138}}$$

Subtract 1 from 5 to get 4.

$$\frac{\left(\sqrt{5}+1\right)\left(\sqrt{5}+1\right)}{4}+\frac{\sqrt{5}-1}{\sqrt{5}}-2\sqrt{\left(\frac{1}{5}\right)^{0}-\frac{1}{138}}$$

Multiply \sqrt{5}+1 and \sqrt{5}+1 to get \left(\sqrt{5}+1\right)^{2}.

$$\frac{\left(\sqrt{5}+1\right)^{2}}{4}+\frac{\sqrt{5}-1}{\sqrt{5}}-2\sqrt{\left(\frac{1}{5}\right)^{0}-\frac{1}{138}}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{5}+1\right)^{2}.

$$\frac{\left(\sqrt{5}\right)^{2}+2\sqrt{5}+1}{4}+\frac{\sqrt{5}-1}{\sqrt{5}}-2\sqrt{\left(\frac{1}{5}\right)^{0}-\frac{1}{138}}$$

The square of \sqrt{5} is 5.

$$\frac{5+2\sqrt{5}+1}{4}+\frac{\sqrt{5}-1}{\sqrt{5}}-2\sqrt{\left(\frac{1}{5}\right)^{0}-\frac{1}{138}}$$

Add 5 and 1 to get 6.

$$\frac{6+2\sqrt{5}}{4}+\frac{\sqrt{5}-1}{\sqrt{5}}-2\sqrt{\left(\frac{1}{5}\right)^{0}-\frac{1}{138}}$$

Rationalize the denominator of \frac{\sqrt{5}-1}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{6+2\sqrt{5}}{4}+\frac{\left(\sqrt{5}-1\right)\sqrt{5}}{\left(\sqrt{5}\right)^{2}}-2\sqrt{\left(\frac{1}{5}\right)^{0}-\frac{1}{138}}$$

The square of \sqrt{5} is 5.

$$\frac{6+2\sqrt{5}}{4}+\frac{\left(\sqrt{5}-1\right)\sqrt{5}}{5}-2\sqrt{\left(\frac{1}{5}\right)^{0}-\frac{1}{138}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4 and 5 is 20. Multiply \frac{6+2\sqrt{5}}{4} times \frac{5}{5}. Multiply \frac{\left(\sqrt{5}-1\right)\sqrt{5}}{5} times \frac{4}{4}.

$$\frac{5\left(6+2\sqrt{5}\right)}{20}+\frac{4\left(\sqrt{5}-1\right)\sqrt{5}}{20}-2\sqrt{\left(\frac{1}{5}\right)^{0}-\frac{1}{138}}$$

Since \frac{5\left(6+2\sqrt{5}\right)}{20} and \frac{4\left(\sqrt{5}-1\right)\sqrt{5}}{20} have the same denominator, add them by adding their numerators.

$$\frac{5\left(6+2\sqrt{5}\right)+4\left(\sqrt{5}-1\right)\sqrt{5}}{20}-2\sqrt{\left(\frac{1}{5}\right)^{0}-\frac{1}{138}}$$

Do the multiplications in 5\left(6+2\sqrt{5}\right)+4\left(\sqrt{5}-1\right)\sqrt{5}.

$$\frac{30+10\sqrt{5}+20-4\sqrt{5}}{20}-2\sqrt{\left(\frac{1}{5}\right)^{0}-\frac{1}{138}}$$

Do the calculations in 30+10\sqrt{5}+20-4\sqrt{5}.

$$\frac{50+6\sqrt{5}}{20}-2\sqrt{\left(\frac{1}{5}\right)^{0}-\frac{1}{138}}$$

Calculate \frac{1}{5} to the power of 0 and get 1.

$$\frac{50+6\sqrt{5}}{20}-2\sqrt{1-\frac{1}{138}}$$

Subtract \frac{1}{138} from 1 to get \frac{137}{138}.

$$\frac{50+6\sqrt{5}}{20}-2\sqrt{\frac{137}{138}}$$

Rewrite the square root of the division \sqrt{\frac{137}{138}} as the division of square roots \frac{\sqrt{137}}{\sqrt{138}}.

$$\frac{50+6\sqrt{5}}{20}-2\times \frac{\sqrt{137}}{\sqrt{138}}$$

Rationalize the denominator of \frac{\sqrt{137}}{\sqrt{138}} by multiplying numerator and denominator by \sqrt{138}.

$$\frac{50+6\sqrt{5}}{20}-2\times \frac{\sqrt{137}\sqrt{138}}{\left(\sqrt{138}\right)^{2}}$$

The square of \sqrt{138} is 138.

$$\frac{50+6\sqrt{5}}{20}-2\times \frac{\sqrt{137}\sqrt{138}}{138}$$

To multiply \sqrt{137} and \sqrt{138}, multiply the numbers under the square root.

$$\frac{50+6\sqrt{5}}{20}-2\times \frac{\sqrt{18906}}{138}$$

Cancel out 138, the greatest common factor in 2 and 138.

$$\frac{50+6\sqrt{5}}{20}-\frac{\sqrt{18906}}{69}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 20 and 69 is 1380. Multiply \frac{50+6\sqrt{5}}{20} times \frac{69}{69}. Multiply \frac{\sqrt{18906}}{69} times \frac{20}{20}.

$$\frac{69\left(50+6\sqrt{5}\right)}{1380}-\frac{20\sqrt{18906}}{1380}$$

Since \frac{69\left(50+6\sqrt{5}\right)}{1380} and \frac{20\sqrt{18906}}{1380} have the same denominator, subtract them by subtracting their numerators.

$$\frac{69\left(50+6\sqrt{5}\right)-20\sqrt{18906}}{1380}$$

Do the multiplications in 69\left(50+6\sqrt{5}\right)-20\sqrt{18906}.

$$\frac{3450+414\sqrt{5}-20\sqrt{18906}}{1380}$$